What were the standard relations for gamma matrices in the mid 50's, when 4-vectors where represented by $(x_1, x_2, x_3, ict)$? In particular the values of $\gamma^\mu\gamma^\nu$ , the definition of $\bar{\psi}$ and $\gamma^5$.


Gregor Wentzel's 1949's book, "Quantum theory of fields" defines the Dirac matrices as

\begin{eqnarray} \alpha ^{(\nu)} &=& \alpha ^{(\nu)*}\\ \alpha ^{(\mu)} \alpha ^{(\nu)} + \alpha ^{(\nu)} \alpha ^{(\mu)} &=& 2 \delta_{\mu\nu} \end{eqnarray}

and the gamma matrices as ($\beta$ is $\alpha^{(4)}$)

\begin{eqnarray} \gamma^{(k)} &=& -i \beta \alpha^{(k)}\\ &=& i \alpha^{(k)} \beta\\ \gamma^{(4)} &=& \beta \end{eqnarray}

with properties

\begin{eqnarray} \gamma^{(\nu)} &=& \gamma^{(\nu)*}\\ \gamma^{(\mu)} \gamma^{(\nu)} + \gamma^{(\nu)} \gamma^{(\mu)} &=& 2 \delta_{\mu\nu} \end{eqnarray}

and the adjoint spinor is

\begin{eqnarray} \psi^\dagger = i \psi^* \beta \end{eqnarray}

Rivier in "On the quantum theory of fields" (1953) defines the matrices by

\begin{eqnarray} \gamma^{i} &=& \beta a^i\\ \gamma^4 &=& \beta \end{eqnarray}


\begin{eqnarray} a^i = \begin{pmatrix} 0 & \sigma^i\\ \sigma^i & 0 \end{pmatrix},\ \beta = \begin{pmatrix} (1) & 0\\ 0 & (1) \end{pmatrix} \end{eqnarray}

with the identities

\begin{eqnarray} [\gamma^\mu, \gamma^\nu]_+ &=& 2 g^{\mu\nu} \gamma^{i+} &=& -\gamma^i\\ \gamma^{4+} &=& \gamma^4 \end{eqnarray}

The adjoint spinor is defined by

\begin{eqnarray} \bar{\psi} = \psi^+_b \beta_{ba} \end{eqnarray}

Morse defines them in Methods of theoretical physics (1953) as

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It's harder to find uses of $\gamma^5$ (most theories were chiral), but you can check Theory of the Fermi interaction (1957), which puts it as $\gamma^5 = \gamma_x \gamma_y \gamma_z \gamma_t$, with

\begin{eqnarray} \gamma^t = \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix},\ \mathbf{\gamma} = \begin{pmatrix} 0 & \mathbf{\sigma}\\ - \mathbf{\sigma} & 0 \end{pmatrix},\ i\gamma^5 = -\begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix} \end{eqnarray}

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